How to cut and paste part of a mesh to another mesh?
You can't do it using the normal copy and paste functions, but you can do it using a few separate shortcuts/tools:

To 'cut' part of a mesh to a new object:

1. In edit mode, select the parts of the mesh you want to cut.

If you want to copy part of the mesh and not cut it you will need to do step '2', otherwise skip to step '3':

2.  Press Shift+D to duplicate the selected faces/edges/verts.

3. Press P and choose Selection from the Separate menu. The selected part of the mesh will now be a separate object but you will still be in edit mode of the original object.

4. Leave edit mode and select the new separate object (which will be on  top of the old object).

If you wish to join this into an existing mesh you can continue:

5. Shift select a different existing mesh and press Ctrl+J to join them together.